English
When ι is finite, multilinear maps on R^ι with values in M₂ are in bijection with M₂ itself via evaluation at the all-ones vector.
Русский
При конечном ι отображения на R^ι с значениями в M₂ эквивариантны самим M₂ через оценку на единичный вектор.
LaTeX
$$There is a linear equivalence πRingEquiv : M₂ ≃ₗ[R] MultilinearMap R (fun _ : ι => R) M₂.$$
Lean4
/-- Two multilinear maps indexed by a `Fintype` are equal if they are equal when all arguments
are basis vectors. -/
theorem ext_multilinear [Finite ι] {f g : MultilinearMap R M N} {ιM : ι → Type*} (e : ∀ i, Basis (ιM i) R (M i))
(h : ∀ v : (i : ι) → ιM i, (f fun i ↦ e i (v i)) = g fun i ↦ e i (v i)) : f = g :=
by
cases nonempty_fintype ι
classical
ext m
rcases Function.Surjective.piMap (fun i ↦ (e i).repr.symm.surjective) m with ⟨x, rfl⟩
unfold Pi.map
simp_rw [(e _).repr_symm_apply, Finsupp.linearCombination_apply, Finsupp.sum, map_sum_finset, map_smul_univ, h]
